We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{ R}^d;\mathbb{ R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled second-order elliptic operators $\bf{ \mathcal A}$, with possibly unbounded coefficients and diffusion and drift terms which vary from equation to equation. We also provide estimates of the spatial derivatives up to the third-order and continuity properties both of the evolution operator ${ \bf G}(t,s)$ associated to the Cauchy problem $D_t{ \bf u}=\bf{ \mathcal A}(t){ \bf u}$ in $C_b(\mathbb{ R}^d;\mathbb{ R}^m)$, and, for fixed $\overline t$, of the semigroup ${ \bf T}_{ \overline t}(\tau)$ associated to the autonomous Cauchy problem $D_{ \tau}{ \bf u}={ \bf{ \mathcal A}}(\overline t){ \bf u}$ in $C_b(\mathbb{ R}^d;\mathbb{ R}^m)$. These results allow us to deal with elliptic problems whose coefficients also depend on time.
Categories: math.AP